摘要
本文在不考慮樑的軸向延展性、質量慣性矩、科氏力（Coriolis force）影響的情況下，利用Hamililton’s principle推導出，一彈性拘束的非均勻、非對稱樑的三條bending-bending-torsion的統御方程式及邊界條件。以半解析方法求得統御特性微分方程式（governing characteristic differential equation）的轉移矩陣（transition matrix），再配合邊界條件而求得系統的頻率方程式，及推導出特徵函數（eigenfunction）的正交條件，再以疊加法得到系統的強迫振動解。本文探討了warping對系統的影響、截面幾何參數、樑長度比例及均勻預扭對自然頻率的影響。

ABSTRACT
The governing differential equations and twelve boundary conditions for the bending-bending-torsion vibration of a nonuniform and asymmetry beam with the elastic boundary conditions are derived by using Hamilton’s principle. The frequency equation of the system is derived and expressed in terms of the transition matrix of the transformed vector characteristic governing equation. An efficient algorithm for determining the semi-analytical transition matrix of the system is derived. The orthogonality condition for the eigenfunctions of the system with elastic boundary conditions is also derived. Using the method of mode superposition derives the analytical solution of force vibration of thin-walled beam. The influence of the warping function, the geometric parameter of the cross-section of the beam, the parameter between the cross-section and length of the beam and the uniform pretwists on the natural frequencies are investigated.

參考文獻
1. S. P. Timoshenko and J. M. Gere., “Theory of Elastic Stability.,” New York : McGraw-Hill, second edition, 1961.
2. V. Z. Vlasov, “Thin-Walled Elastic Beams,” Jerusalem : Israel Program for Scientific Translation, 1961.
3. S. P. Timoshenko, ”Theory of Bending, Torsion and Buckling of Thin-Walled Members of Open Section,” Journal of Franklin Institute, Vol. 239, 1945, pp. 215-219, 249-254.
4. J. M. Gere, “Torsional Vibrations of Beams of Thin-Walled Open Section.,” Journal of Applied Mechanics, Vol. 21, 1954, pp. 381-387.
5. J. M. Gere and Y. K. Lin, “Coupled Vibrations of Thin-Walled Beams of Open Cross Section,” Journal of Applied Mechanics, Vol. 25, 1958, pp. 373-378.
6. Y. K. Lin, “Coupled Bending and Torsional Vibrations of Restrained Thin-Walled Beams,” Journal of Applied Mechanics, Vol. 27, 1960, pp. 739-740.
7. H. R. Aggarwal and E. T. Cranch, “A Theory of Torsional and Coupled Bending Torsional Waves in Thin-Walled Open Section Beams,” Journal of Applied Mechanics, Vol. 34, 1967, pp. 337-343.
8. Y. Y. Yu, “Variational Equation of Motion for Coupled Flexure and Torsion of Bars of Thin-Walled Open Section Including Thermal Effect,” Journal of Applied Mechanics, Vol. 38, 1971, pp. 502-506.
9. S. Dubigeon and C. B. Kim, “A Finite Element for The Study of Coupled Bending-Prevented Torsion of a Straight Beam,” Journal of Applied Mechanics, Vol. 81, 1982, pp. 255-270.
10. A. W. Leissa, “Vibrations of Shells,” NASA SP-288, 1973.
11. T. Irie, G. Yamada and K. Tanaka, “Free Vibration of A Thin-Walled Beam-Shell of Arc Cross-Section,” Journal of Sound and Vibration, Vol. 94, No 4,1984, pp. 563-572.
12. G. A. Gunnlaugsson and P. T. Pederson, “A Finite Element Formulation for Beams with Thin Walled Cross-Sections,” Comput. Struct. Vol. 15, 1982, pp. 691-699.
13. T. Kawai, “The Application of Finite Element Methods to Ship Structures,” Comput. Struct. Vol. 3, 1973, pp. 1175-1194.
14. I. Senjanovic and R. Grubisic, “Coupled Horizontal and Torsional Vibration of Ship Hull with Large deck opening,” Numerical Analysis of The Dynamics of Ship Structures, Vol. 122, 1979, pp. 539-556.
15. R. S. Barsoum and R. H. Gallagher, “Finite Element Analysis of Torsional and Torsional Flexural Stability Problems,” Int. J. Numer. Meth. Engng, Vol. 2, 1970, pp. 335-353.
16. J. L. Meek and P. Swannel, “Stiffness Matrices for Beam Members Including Warping Torsion Effect,” J. Engng Mech. Div., ASCE, 1976, pp. 193-197.
17. A. Pittaluga, “Recent Developments in The Theory of Thin Walled Beams,” Computer & Structures, N0. 3, 1978, pp. 69-79.
18. K. J. Bathe, “Finite Element Procedures in Engineering Analysis,” Prentice-Hall, Englewood Cliffs, NJ, 1982.
19. R. D. Cook, “Concepts and Applications of Finite Element Analysis,” 2nd Edn. John Wiley, New York, 1982.
20. A. K. Noor and J. M. Peters, “Mixed Model and Reduced/Selective Integration Displacement Models for Nonlinear Analysis of Curved Beams,” Int. J. Numer. Meth. Engng, Vol. 17, 1981, pp. 615-631.
21. H. Stolarski and T. Belytschko, “Shear and Membrane Locking in 0 Curved C Elements,” Comput. Meth. Appl. Mech. Engng, Vol. 41, 1983, pp. 172-176.
22. Erkan Dokumaci, “Exact Analysis of Coupled Bending Torsion Vibrations of Asymmetric Beams,” The American Society of Mechanical Engineers, Winter Annual Meeting, 1985.
23. C. P. Filipich and M. B. Rosales, “Free Flexural-Torsional Vibrations of A Uniform Spinning Beam,” Journal of Sound and Vibration, Vol. 141, No.3, 1990, pp. 375-387.
24. Dawson, B., “Coupled Bending Vibrations of Pretwisted Cantilever Blading Treated by Rayleigh-Ritz Method,” Journal of Mechanical Engineering Science, Vol. 10, No. 5, 1968, pp. 381-386.
25. Dawson, B., and Carnegie, W., “Modal Curves of Pre-Twisted Beams of Rectangular Cross-Section,” Journal of Mechanical Engineering Science, Vol. 11, No. 1, 1969, pp. 1-13.
26. Carnegie, W., and Thomas, J., “The Coupled Bending-Bending Vibration of Pre-Twisted Tapered Blading,” Journal of engineering Industry, Vol. 94, Feb. 1972, pp. 255-266.
27. Rao, J. S., “Flexural Vibration of Pretwisted Tapered Cantilever Blades,” Journal of Engineering Industry, Vol. 94, Feb. 1972, pp. 343-346.
28. Rao, J. S., “Coupled Vibrations of Turbomachine Blading,” Shock and Vibration Bulletin, Vol. 47, Pt. 2, 1977, pp. 107-125.
29. Rao, J. S., Advanced Theory of Vibration, Wiley, New York, 1992, pp. 330-338.
30. Rao, J. S., and Carnegie, W., “A Numerical Procedure for the Determin- ation of the Frequencies and Mode Shapes of Lateral Vibration of Blades Allowing for the Effect of Pre-Twist and Rotation,” International Journal of Mechanical Engineering Eduacation, Vol. 1, No. 1, 1973, pp. 37-47.
31. Lin, S. M., “Vibrations of Elastically Restrained Nonuniform Beams with Arbitrary Pretwist,” AIAA Journal, Vol. 35, No. 11, Nov. 1997, pp. 1681-1687.
32. Subrabmanyam, K. B., and Kaza, K. R. V., “Vibration and Buckling of Rotating, Pretwisted, Preconed Beams Including Coriolis Effects,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, No. 2, 1986, pp. 140-149.
33. Subrabmanyam, K. B., Kulkarni, S. V., and Rao, J. S., “Analysis of Lateral Vibrations of Rotating Cantilever Blades Allowing for Shear Deflection and Rotary Inertia by Reissner and Potential Energy Methods,” Mech Mach Theory, Vol. 17, No. 4, 1982, pp. 235-241.
34. Sisto, F., and Chang, A. T., “Finite Element for Vibration Analysis of Twisted Blades Based on Beam Theory,” AIAA Journal, Vol. 22, No. 11, 1984, pp. 1646-1651.
35. Hernried, A. G., “Forced Vibration Response of a Twisted Non-Uniform Rotating Blade,” Computers and Structures, Vol.41, No. 2, 1991, pp.207-212.
36. S. M. Lin, “The Instability and Vibration of Rotating Beams with Arbitrary Pretwist and An Elastically Restrained Root,” ASME Journal of Applied Mechanics, Vol. 68, 2001, pp. 844-853.